Recent Changes to GSI 3D-Var Analysis

1. Recent Changes to GSI 3D-Var Analysis Wan-Shu Wu March 2007 at CWB

2. Adaptive Tuning of Observational Error

3. Analysis system produces an analysis through the minimization of an objective function given byJ = xT B-1 x + ( H x – y )T R-1( H x – y) Jb Jo Where x is a vector of analysis increments, B is the background error covariance matrix, y is a vector of the observational residuals, y = y obs – H xguess R is the observational and representativeness error covariance matrix H is the transformation operator from the analysis variable to the form of the observations.

4. Talagrand (1997, 1999) on E ( J (Xa) ) • Desroziers & Ivanov (2001): Application • The expectation of J’s at the minima are E( Jo )= ½ Tr ( Ip – HK) E( Jb )= ½ Tr (KH) where Ip is identity matrix with order p K is Kalman gain matrix H is linearlized observation forward operator • Chapnik et al.(2004): robust even when B is incorrectly specified

5. Adaptive tuning procedure J ( dX) =1/sb2 Jb ( dX) + 1/so2 Jo ( dX) Where sb and so are the background and oberr weighting parameters So= sqrt( 2Jo / Tr ( Ip – HK) )

6. Randomized estimation of Tr (HK) • Tr ( Ip – HK) = Nobs - ( Sx R-1/2 H dXa(y+R1/2x) - Sx R-1/2 H dXa(y) ) where x is random number with standard Gaussian distribution ( mean: 0 variance:1 ) • 2 outer iterations to produce each adaptive tuning; output new error table • Consecutive jobs show the method converged; sum decreases with each iteration Sum = S (1-so)2

7. So for each ob type • So function of height (pressure) rawinsonde, aircft, aircar, profiler winds, dropsonde, satwind… • So constant with height ship, synoptic, metar, bogus, ssm/I, ers speed, aircft wind, aircar wind…

11. Tuning results Wind oberrtmp oberr Ob type 180-189

12. Impact of oberr tuning on forecasts3 hr forecast RMS fit to obs at end of 12-hr cycle (cntl/exp)blue : positive: red : negative

13. Normalized Relative Humidity* *Holm et al.(2002) ECMWF Tech Memo 383

14. Requirements on humidity as analysis control variable • Non negative & limited super-saturation • Gaussian standard deviation with zero bias • Example of humidity analysis variables q: specific humidity log q: logarithmic of specific humidity RH: relative humidity Pseudo relative humidity …

15. Characteristics of humidity variables

16. Construction of a Gaussian control variable

17. Construction of a Gaussian control variable

18. Normalized relative humidity as analysis control variable* RH(p,q,t) = pq / (ges (T) ) dRH = RHb ( dP/Pb + dq /qb + dT /ab ) Normalized RH= dRH / s(RHb) where s(RHb) : standard deviation of B as function of RHb ab : 1 / d(RH)/d(T) Note: besides humidity, multivariate relation with T and P *Holm et al.(2002) ECMWF Tech Memo

19. Linearized Multivariate relation Note: The inverse of the matrix which defines the new analysis variable is required where as only the adjoint of the observational forward operator is needed in 3D Var.

20. The 2 options produce similar analysis increments at z=1, NMM central 8 domain

21. Vertical sections x=330 : similar analysis increments in the lower atmosphere

22. At z=60 (top of the model) Option 1 extends the RH forcing from below Option 2 keeps zero RH forcing with T and q multivariate relation.

23. Scales of q1 & q2

24. Positive impact on wind through 84 hr forecasts. Consistent with the cycling results

25. Positive impact on temp in first guess (3hr forecasts) in NDAS. The advantage last through 12 hr forecasts but become neutral by 84 hours.

26. Strong constraint balance method in GSI David Parrish, Daryl Kleist, Ron Errico, Runhua Yang NOAA/NWS/NCEP/EMC

27. Weak constraint GSI J = Jb + Jo + Jc J = (x-xb)TB-1(x-xb) + (H(x)-y0)T(E+F)-1(H(x)-y0) + JC J = Fit to background + Fit to observations + constraints x = Analysis xb = Background B = Background error covariance H = Forward model y0 = Observations E+F = R = Instrument error + Representativeness error JC = Weak constraint term

28. Problems with JC • Poor convergence of analysis iterations • Significant degradation of fit to observations • Large negative impact in first test with NAM.

29. New method J = Jb + Jo(remove weak balance constraint JC) J = (x-xb)TB-1(x-xb) + (H(w)-y0)T(E+F)-1(H(w)-y0) J = Fit to background + Fit to observations w = xb + N(x) – N(xb) N = nonlinear normal mode initialization (NNMI) w = analysis state vector after incremental NNMI

30. Advantages of new method • By adding NNMI to forward model, analysis increment is always in good dynamic balance. • Convergence of analysis still good • Ability to adjust background/obs errors over wider range to tune for optimal performance • Same cost as weak constraint JC

31. Two versions of strong constraintin GSI • Global version – based on Temperton, C., 1989: “Implicit Normal Mode Initialization for Spectral Models”, MWR, vol 117, 436-451. • Regional version – based on Briere, S., 1982: “Nonlinear Normal Mode Initialization of a Limited Area Model”, MWR, vol 110, 1166-1186.

32. Summary • Adaptive tuning of the observational error • Normalized relative humidity as analysis variable • Strong constraint • Use pressure instead of log(p) as analysis variable • Use CRTM in satellite data assimilation • Use ensemble method to estimate B • MIP_IO capability • Use of GPS local refractivity and bending angle • Variational / non-linear QC